4. Volumes by
Cylindrical Shells
APPLICATIONS OF INTEGRATION
In this section, we will learn:
How to apply the method of cylindrical shells
to find out the volume of a solid.
5. Let’s consider the problem of finding the
volume of the solid obtained by rotating about
the y-axis the region bounded by y = 2x2
- x3
and y = 0.
VOLUMES BY CYLINDRICAL SHELLS
6. If we slice perpendicular to the y-axis,
we get a washer.
However, to compute the inner radius and the outer
radius of the washer,
we would have to
solve the cubic
equation y = 2x2
- x3
for x in terms of y.
That’s not easy.
VOLUMES BY CYLINDRICAL SHELLS
7. Fortunately, there is a method—the
method of cylindrical shells—that is
easier to use in such a case.
VOLUMES BY CYLINDRICAL SHELLS
8. The figure shows a cylindrical shell
with inner radius r1, outer radius r2,
and height h.
CYLINDRICAL SHELLS METHOD
9. Its volume V is calculated by subtracting
the volume V1 of the inner cylinder from
the volume of the outer cylinder V2 .
CYLINDRICAL SHELLS METHOD
10. Thus, we have:
2 1
2 2
2 1
2 2
2 1
2 1 2 1
2 1
2 1
( )
( )( )
2 ( )
2
V V V
r h r h
r r h
r r r r h
r r
h r r
π π
π
π
π
= −
= −
= −
= + −
+
= −
CYLINDRICAL SHELLS METHOD
11. Let ∆r = r2 – r1 (thickness of the shell) and
(average radius of the shell).
Then, this formula for the volume of a
cylindrical shell becomes:
2V rh rπ= ∆
Formula 1
( )1
2 12r r r= +
CYLINDRICAL SHELLS METHOD
12. The equation can be remembered as:
V = [circumference] [height] [thickness]
CYLINDRICAL SHELLS METHOD
2V rh rπ= ∆
13. Now, let S be the solid
obtained by rotating
about the y-axis the
region bounded by
y = f(x) [where f(x) ≥ 0],
y = 0, x = a and x = b,
where b > a ≥ 0.
CYLINDRICAL SHELLS METHOD
14. Divide the interval [a, b] into n subintervals
[xi - 1, xi ] of equal width and let be
the midpoint of the i th subinterval.
x∆ix
CYLINDRICAL SHELLS METHOD
15. The rectangle with
base [xi - 1, xi ] and
height is rotated
about the y-axis.
The result is a
cylindrical shell with
average radius ,
height , and
thickness ∆x.
( )if x
( )if x
ix
CYLINDRICAL SHELLS METHOD
16. Thus, by Formula 1, its volume is
calculated as follows:
(2 )[ ( )]i i iV x f x xπ= ∆
CYLINDRICAL SHELLS METHOD
17. So, an approximation to the volume V of S
is given by the sum of the volumes of
these shells:
1 1
2 ( )
n n
i i i
i i
V V x f x xπ
= =
≈ = ∆∑ ∑
CYLINDRICAL SHELLS METHOD
18. The approximation appears to become better
as n →∞.
However, from the definition of an integral,
we know that:
1
lim 2 ( ) 2 ( )
n b
i i an
i
x f x x x f x dxπ π
→∞
=
∆ =∑ ∫
CYLINDRICAL SHELLS METHOD
19. Thus, the following appears plausible.
The volume of the solid obtained by rotating
about the y-axis the region under the curve
y = f(x) from a to b, is:
where 0 ≤ a < b
2 ( )
b
a
V xf x dxπ= ∫
Formula 2CYLINDRICAL SHELLS METHOD
20. The argument using cylindrical shells
makes Formula 2 seem reasonable,
but later we will be able to prove it.
CYLINDRICAL SHELLS METHOD
21. Here’s the best way to remember
the formula.
Think of a typical shell,
cut and flattened,
with radius x,
circumference 2πx,
height f(x), and
thickness ∆x or dx:
( ){
[ ] {2 ( )
b
a
thicknesscircumference height
x f x dxπ∫ 123
CYLINDRICAL SHELLS METHOD
22. This type of reasoning will be helpful
in other situations—such as when we
rotate about lines other than the y-axis.
CYLINDRICAL SHELLS METHOD
23. Find the volume of the solid obtained by
rotating about the y-axis the region
bounded by y = 2x2
- x3
and y = 0.
Example 1CYLINDRICAL SHELLS METHOD
24. We see that a typical shell has
radius x, circumference 2πx, and
height f(x) = 2x2
- x3
.
Example 1CYLINDRICAL SHELLS METHOD
25. So, by the shell method,
the volume is:
( ) ( )
( )
( )
2
2 3
0
2
3 4
0
24 51 1
2 5 0
32 16
5 5
2 2
2 (2 )
2
2 8
π
π
π
π π
= −
= −
= −
= − =
∫
∫
V x x x dx
x x x dx
x x
Example 1CYLINDRICAL SHELLS METHOD
26. It can be verified that the shell method
gives the same answer as slicing.
The figure shows
a computer-generated
picture of the solid
whose volume we
computed in the
example.
Example 1CYLINDRICAL SHELLS METHOD
27. Comparing the solution of Example 1 with
the remarks at the beginning of the section,
we see that the cylindrical shells method
is much easier than the washer method
for the problem.
We did not have to find the coordinates of the local
maximum.
We did not have to solve the equation of the curve
for x in terms of y.
NOTE
28. However, in other examples,
the methods learned in Section 6.2
may be easier.
NOTE
29. Find the volume of the solid obtained
by rotating about the y-axis the region
between y = x and y = x2
.
Example 2CYLINDRICAL SHELLS METHOD
30. The region and a typical shell
are shown here.
We see that the shell has radius x, circumference 2πx,
and height x - x2
.
Example 2CYLINDRICAL SHELLS METHOD
31. Thus, the volume of the solid is:
( ) ( )
( )
1
2
0
1
2 3
0
13 4
0
2
2
2
3 4 6
V x x x dx
x x dx
x x
π
π
π
π
= −
= −
= − =
∫
∫
Example 2CYLINDRICAL SHELLS METHOD
32. As the following example shows,
the shell method works just as well
if we rotate about the x-axis.
We simply have to draw a diagram to identify
the radius and height of a shell.
CYLINDRICAL SHELLS METHOD
33. Use cylindrical shells to find the volume of
the solid obtained by rotating about the x-axis
the region under the curve from 0 to
1.
This problem was solved using disks in Example 2
in Section 6.2
y x=
Example 3CYLINDRICAL SHELLS METHOD
34. To use shells, we relabel the curve
as x = y2
.
For rotation about
the x-axis, we see that
a typical shell has
radius y, circumference
2πy, and height 1 - y2
.
y x=
Example 3CYLINDRICAL SHELLS METHOD
35. So, the volume is:
In this problem, the disk method was simpler.
( ) ( )
1
2
0
1
3
0
12 4
0
2 1
2 ( )
2
2 4 2
V y y dy
y y dy
y y
π
π
π
π
= −
= −
= − =
∫
∫
Example 3CYLINDRICAL SHELLS METHOD
36. Find the volume of the solid obtained by
rotating the region bounded by y = x - x2
and y = 0 about the line x = 2.
Example 4CYLINDRICAL SHELLS METHOD
37. The figures show the region and a cylindrical
shell formed by rotation about the line x = 2,
which has radius 2 - x, circumference
2π(2 - x), and height x - x2
.
Example 4CYLINDRICAL SHELLS METHOD
38. So, the volume of the solid is:
( ) ( )
( )
0
2
1
0
3 2
1
14
3 2
0
2 2
2 3 2
2
4 2
V x x x dx
x x x dx
x
x x
π
π
π
π
= − −
= − +
= − + =
∫
∫
Example 4CYLINDRICAL SHELLS METHOD
42. Washers
Consider the area between two functions
rotated about the axis
Now we have a hollow solid
f(x)
a b
g(x
)
43. Washers
Consider the area between two functions
rotated about the axis
Now we have a hollow solid
We will sum the volumes of washers
f(x)
a b
g(x
)
45. Find the volume of the solid formed by revolving the
region bounded by y = √(x) and y = x² over the interval
[0, 1] about the x – axis.
46. Find the volume of the solid formed by revolving the
region bounded by y = √(x) and y = x² over the interval
[0, 1] about the x – axis.
2 2
([ ( )] [ ( )] )
b
a
V f x g x dxπ= −∫
47. Find the volume of the solid formed by revolving the
region bounded by y = √(x) and y = x² over the interval
[0, 1] about the x – axis.
2 2
([ ( )] [ ( )] )
b
a
V f x g x dxπ= −∫
( ) ( )∫ −=
1
0
222
dxxxV π
48. Find the volume of the solid formed by revolving the
region bounded by y = √(x) and y = x² over the interval
[0, 1] about the x – axis.
2 2
([ ( )] [ ( )] )
b
a
V f x g x dxπ= −∫
( ) ( )∫ −=
1
0
222
dxxxV π
( )∫ −=
1
0
4
dxxxV π
49. Find the volume of the solid formed by revolving the
region bounded by y = √(x) and y = x² over the interval
[0, 1] about the x – axis.
2 2
([ ( )] [ ( )] )
b
a
V f x g x dxπ= −∫
( ) ( )∫ −=
1
0
222
dxxxV π
( )∫ −=
1
0
4
dxxxV π
1
0
52
5
1
2
1
−= xxV π
50. Find the volume of the solid formed by revolving the
region bounded by y = √(x) and y = x² over the interval
[0, 1] about the x – axis.
2 2
([ ( )] [ ( )] )
b
a
V f x g x dxπ= −∫
( ) ( )∫ −=
1
0
222
dxxxV π
( )∫ −=
1
0
4
dxxxV π
1
0
52
5
1
2
1
−= xxV π
10
3π
=